Hi guys, I'm a lil confused on subject of equality in math ..
lets assume that there is a condition that fulfills the following equation : i^2<n ;
then it's equal to say that the condition itself fulfills in other words i<sqrt(n), but why? why it's the same to say that thing if I have changed the prime condition?

may anyone please explain me the logic of equinumerosity in math?

 

in other words i<sqrt(n), but why?

Because the square-root of a number is always smaller than the number.

 

Because the square-root of a number is always smaller than the number.

You don't understand me !
I mean why if the condition fulfills the i^2<n then it's the same to say that the condition fulfills i<sqrt(n)

 

Hi guys, I'm a lil confused on subject of equality in math ..
lets assume that there is a condition that fulfills the following equation : i^2<n ;
then it's equal to say that the condition itself fulfills in other words i<sqrt(n), but why? why it's the same to say that thing if I have changed the prime condition?

may anyone please explain me the logic of equinumerosity in math?

First you should clean up your vocabulary: equality is not the same thing as equinumerosity, and neither are the same thing as the '<' relation. In short, when two things are equinumerous, they contain the same number of things (typically this applies to the cardinality of finite sets). Equality is trickier to pin down, as it means different things in different mathematical contexts. But the usual interpretation is that when two things are equal, they represent the same object. Finally, '<' is a binary relation that says that when two things can be ordered, the thing on the left is less than the thing on the right.

As for your example, it does not follow that if i^2 < n, then i < sqrt(n). First, you should explicitly state the domain of discourse: are we talking about real numbers, complex numbers, integers mod 2? Note that square roots are not always defined to exist, so the domain is important. Second, if we assume that sqrt(n) exists, then we have to treat it as multivalued. In particular, -sqrt(n) will be a solution [because (-n)^2 = n]. Since sqrt(i^2) is positive by definition (equal to the absolute value of i), your conclusion is false.

Here's a counterexample over the integers: let i = 2 and n = 100. Then, 2 < -10 is a valid (though false) solution to i < sqrt(n).

 

I mean why if the condition fulfills the i^2<n then it's the same to say that the condition fulfills i<sqrt(n)

Sorry, I misread.
If you do the same mathematical operation to both sides of the function, it does not change the function.
So taking the square-root of both sides of the function does not change the inequality.
Thus if you take the square-root of both sides of (i²<n) you get (√i² < √n) or (i < √n)

 

Because the square-root of a number is always smaller than the number.

Really?

So sqrt(0.5) < 0.5

 

You don't understand me !
I mean why if the condition fulfills the i^2<n then it's the same to say that the condition fulfills i<sqrt(n)

First off, you seem to like to use unusual words in places where they don't really belong. You might try to stop doing that since it just serves to confuse everyone.

The concept of "equinumerosity" has to do with the size of sets (as in "equally numerous" or "having the same number of elements"). It has nothing to do with the equality of two expressions.

In general, if two expressions are equal, then performing the same operation to both sides maintains the equality.

There are some caveats here, namely were relations are multivalues. The sqrt() is a prime example. The sqrt(9) can be either +3 or -3. Often times we assume that we are only concerned with the principal square root (the +3), but in many instances we need to allow for either case and explore all cases carefully.

Another huge caveat applies to inequalities because if you do anything that inverts the expressions (such as multiplying by a negative value), then the sense of the inequality gets flipped. Tracking this is something that can get quite tricky and needs to be done with care.

 

First you should clean up your vocabulary: equality is not the same thing as equinumerosity, and neither are the same thing as the '<' relation. In short, when two things are equinumerous, they contain the same number of things (typically this applies to the cardinality of finite sets).

FWIW, it is also applied to infinite sets and it is probably there that it has its greatest impact.

 

FWIW, it is also applied to infinite sets and it is probably there that it has its greatest impact.

I could be wrong, but I don't believe mathematicians would describe, say, ℤ and ℕ as equinumerous, even though they have the same cardinality. In fact, I believe that's why the notion of cardinality (bijection between sets) was invented in the first place: feels wrong to say that there are "as many" even natural numbers as their are natural numbers, when clearly there are "twice as many"!

 

First off, you seem to like to use unusual words in places where they don't really belong. You might try to stop doing that since it just serves to confuse everyone.

The concept of "equinumerosity" has to do with the size of sets (as in "equally numerous" or "having the same number of elements"). It has nothing to do with the equality of two expressions.

In general, if two expressions are equal, then performing the same operation to both sides maintains the equality.

There are some caveats here, namely were relations are multivalues. The sqrt() is a prime example. The sqrt(9) can be either +3 or -3. Often times we assume that we are only concerned with the principal square root (the +3), but in many instances we need to allow for either case and explore all cases carefully.

Another huge caveat applies to inequalities because if you do anything that inverts the expressions (such as multiplying by a negative value), then the sense of the inequality gets flipped. Tracking this is something that can get quite tricky and needs to be done with care.

well, firstly I didn't intend to use this terms in wrong places .. I just actually was thinking it's the same term as "equal" ..anyway thanks for your notes !

secondly, still not understanding the subject .. frankly !
lets say I have specific statement that apply this equation: x^2=16; why is it equal to say it's the same to say that condition apply this equation: x=4, x=-4 ?! once again I'm not meaning how did we find x=4,x=-4 , I'm asking why is it the same to say either x^2=16 or (x=-4,x=4) ? when we determine that two equations are "equal" that we can exchange between them because they are equal which leads to be the same "equation" ?!!

 

I'm asking why is it the same to say either x^2=16 or (x=-4,x=4) ? when we determine that two equations are "equal" that we can exchange between them because they are equal which leads to be the same "equation" ?!!

The '=' symbol in "x^2 = 16" does not mean equality (i.e., that the thing on the left is the same as the thing on the right). In this context, the '=' symbol means that the expression is an equation, i.e., for some domain D there may or may not be solutions x ∈ D that make the equation true. In the domain of integers, for instance, there are two such solutions: x = -4 and x = 4. We can replace the 'x' symbol in the equation with either '-4' or '4' and the equation is true.

Note that the equation "x^2 = 16" is not the same mathematical object as the set {-4, 4}, or the expressions "x ∈ {-4 , 4}" or "x = -4, x = 4". None of those things are equal. Rather, we say that "x ∈ {-4, 4}" satisfies the equation "x^2 = 16".

 

So sqrt(0.5) < 0.5

Opps.
Obviously I was thinking about positive integers.

 

Opps.
Obviously I was thinking about positive integers.

I started an answer earlier but quickly started thinking about fractions, imaginary numbers, multiple roots, and the inversion problem mentioned above by @WBahn. Since I didn’t really grasp what the TS was asking in the first place, I walked away. Just too much work.

 

The '=' symbol in "x^2 = 16" does not mean equality (i.e., that the thing on the left is the same as the thing on the right). In this context, the '=' symbol means that the expression is an equation, i.e., for some domain D there may or may not be solutions x ∈ D that make the equation true. In the domain of integers, for instance, there are two such solutions: x = -4 and x = 4. We can replace the 'x' symbol in the equation with either '-4' or '4' and the equation is true.

Note that the equation "x^2 = 16" is not the same mathematical object as the set {-4, 4}, or the expressions "x ∈ {-4 , 4}" or "x = -4, x = 4". None of those things are equal. Rather, we say that "x ∈ {-4, 4}" satisfies the equation "x^2 = 16".

I got the point ; but because I look at things as what it " looks/represented" find it difficult to believe that to say x^2=16 is the same to say x=4,-4

 

I started an answer earlier but quickly started thinking about fractions, imaginary numbers, multiple roots, and the inversion problem mentioned above by @WBahn. Since I didn’t really grasp what the TS was asking in the first place, I walked away. Just too much work.

Thanks very much

 

I got the point ; but because I look at things as what it " looks/represented" find it difficult to believe that to say x^2=16 is the same to say x=4,-4

Math notation isn't always perfectly clear, but you get used to it.

 

I could be wrong, but I don't believe mathematicians would describe, say, ℤ and ℕ as equinumerous, even though they have the same cardinality. In fact, I believe that's why the notion of cardinality (bijection between sets) was invented in the first place: feels wrong to say that there are "as many" even natural numbers as their are natural numbers, when clearly there are "twice as many"!

Doesn't really matter if it "feels wrong". When discussing infinite sets, lots of things don't "feel right" to our everyday sense of propriety.

If I can set up a one-to-one correspondence between two sets, finite or infinite, such that each member of one set is paired with a unique and distinct member of the other set, then the sets are defined to be of the same size (and this is very much in accordance with our intuitive sense of how things must be).

But there's no need to take my word for this.

https://en.wikipedia.org/wiki/Equinumerosity

And let your fingers do the searching and you will find no shortage of even more reliable sites that make it clear that two sets are equinumerous if they have the same cardinality -- keep in mind that these are not synonyms; equinumerous is a relationship between two sets while cardinality is the measure of the size of a single set.

 

well, firstly I didn't intend to use this terms in wrong places .. I just actually was thinking it's the same term as "equal" ..anyway thanks for your notes !

secondly, still not understanding the subject .. frankly !
lets say I have specific statement that apply this equation: x^2=16; why is it equal to say it's the same to say that condition apply this equation: x=4, x=-4 ?! once again I'm not meaning how did we find x=4,x=-4 , I'm asking why is it the same to say either x^2=16 or (x=-4,x=4) ? when we determine that two equations are "equal" that we can exchange between them because they are equal which leads to be the same "equation" ?!!

I'm still not following you terminology.

What are we "exchanging between them" and how is that leading to the same equation?

If I have an equation

x + 9 = y + 13

All I am saying is that the left side is equal to the right side.

In order for this to be true, I can't just pick any x and any y, but only those pairs of (x,y) that satisfy this equation, meaning only that when I plug in x on the left and y on the right, that the left evaluates to the same value that the right does.

Now let's say that subtract 7 from both sides and then square both sides.

x² + 4x + 4 = y² + 12y + 36

This is a different equation. But because I arrived at it by applying the same operations to both sides, I'm guaranteed that whatever points (x,y) satisfy the first equation will also satisfy this equation.

But I may also now have additional points that satisfy the equation. For instance, the original equation was satisfied by the points (0,-4) and (4,0) -- in fact all points in which the x value is exactly four greater than the y value.

But now plug in y=0 into the second equation and you get

x² + 4x - 32 = 0

As expected, x = 4 is a solution, but so is x = -8.

I think you need to sit down and spend some serious quality time with a good introductory algebra text and work your way through it step by step.

 

I got the point ; but because I look at things as what it " looks/represented" find it difficult to believe that to say x^2=16 is the same to say x=4,-4

I don't think anyone is saying that it is saying the same thing.

Saying that x² = 16 merely imposes constraints on the values that x can take on and have this equation satisfied. If it takes on either x = 4 or x = -4, then the equation will be satisfied. If it takes on other values, the equation won't be satisfied.

 

Opps.
Obviously I was thinking about positive integers.

Easy mistake to make.

Aside: A student is a member of the Square Root Club if taking the square root of their GPA yields a higher number.